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Theorem cxploglim 24924

Description: The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)

**Assertion**
Ref	Expression
cxploglim	⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Distinct variable group: 𝐴,𝑛

Proof of Theorem cxploglim Dummy variables 𝑚 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpre 12041	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		reefcl 15022	. . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘𝐴) ∈ ℝ)
4		efgt1 15051	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 1 < (exp‘𝐴))
5		cxp2limlem 24922	. . 3 ⊢ (((exp‘𝐴) ∈ ℝ ∧ 1 < (exp‘𝐴)) → (𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0)
6	3, 4, 5	syl2anc 565	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0)
7		reefcl 15022	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ → (exp‘𝑧) ∈ ℝ)
8	7	adantl 467	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → (exp‘𝑧) ∈ ℝ)
9		1re 10240	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ifcl 4267	. . . . . . 7 ⊢ (((exp‘𝑧) ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) ∈ ℝ)
11	8, 9, 10	sylancl 566	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) ∈ ℝ)
12	9	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → 1 ∈ ℝ)
13	8	adantr 466	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → (exp‘𝑧) ∈ ℝ)
14		rpre 12041	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
15	14	adantl 467	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → 𝑛 ∈ ℝ)
16		maxlt 12228	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (exp‘𝑧) ∈ ℝ ∧ 𝑛 ∈ ℝ) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 ↔ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛)))
17	12, 13, 15, 16	syl3anc 1475	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 ↔ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛)))
18		simprrr 759	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝑧) < 𝑛)
19		reeflog 24547	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (exp‘(log‘𝑛)) = 𝑛)
20	19	ad2antrl 699	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘(log‘𝑛)) = 𝑛)
21	18, 20	breqtrrd 4812	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝑧) < (exp‘(log‘𝑛)))
22		simplr 744	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑧 ∈ ℝ)
23	14	ad2antrl 699	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ∈ ℝ)
24		simprrl 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 1 < 𝑛)
25	23, 24	rplogcld 24595	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ⁺)
26	25	rpred 12074	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ)
27		eflt 15052	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℝ ∧ (log‘𝑛) ∈ ℝ) → (𝑧 < (log‘𝑛) ↔ (exp‘𝑧) < (exp‘(log‘𝑛))))
28	22, 26, 27	syl2anc 565	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (𝑧 < (log‘𝑛) ↔ (exp‘𝑧) < (exp‘(log‘𝑛))))
29	21, 28	mpbird 247	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑧 < (log‘𝑛))
30		breq2 4788	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (log‘𝑛) → (𝑧 < 𝑚 ↔ 𝑧 < (log‘𝑛)))
31		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → 𝑚 = (log‘𝑛))
32		oveq2 6800	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → ((exp‘𝐴)↑_𝑐𝑚) = ((exp‘𝐴)↑_𝑐(log‘𝑛)))
33	31, 32	oveq12d 6810	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (log‘𝑛) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) = ((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛))))
34	33	fveq2d 6336	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (log‘𝑛) → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) = (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))))
35	34	breq1d 4794	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (log‘𝑛) → ((abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥 ↔ (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥))
36	30, 35	imbi12d 333	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (log‘𝑛) → ((𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) ↔ (𝑧 < (log‘𝑛) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥)))
37	36	rspcv 3454	. . . . . . . . . . . . 13 ⊢ ((log‘𝑛) ∈ ℝ⁺ → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (𝑧 < (log‘𝑛) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥)))
38	25, 37	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (𝑧 < (log‘𝑛) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥)))
39	29, 38	mpid 44	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥))
40	1	ad2antrr 697	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝐴 ∈ ℝ)
41	40	relogefd 24594	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘(exp‘𝐴)) = 𝐴)
42	41	oveq2d 6808	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = ((log‘𝑛) · 𝐴))
43	25	rpcnd 12076	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℂ)
44		rpcn 12043	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
45	44	ad2antrr 697	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝐴 ∈ ℂ)
46	43, 45	mulcomd 10262	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · 𝐴) = (𝐴 · (log‘𝑛)))
47	42, 46	eqtrd 2804	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = (𝐴 · (log‘𝑛)))
48	47	fveq2d 6336	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘((log‘𝑛) · (log‘(exp‘𝐴)))) = (exp‘(𝐴 · (log‘𝑛))))
49	3	ad2antrr 697	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ∈ ℝ)
50	49	recnd 10269	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ∈ ℂ)
51		efne0 15032	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0)
52	45, 51	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ≠ 0)
53	50, 52, 43	cxpefd 24678	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((exp‘𝐴)↑_𝑐(log‘𝑛)) = (exp‘((log‘𝑛) · (log‘(exp‘𝐴)))))
54		rpcn 12043	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℂ)
55	54	ad2antrl 699	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ∈ ℂ)
56		rpne0 12050	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≠ 0)
57	56	ad2antrl 699	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ≠ 0)
58	55, 57, 45	cxpefd 24678	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (𝑛↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑛))))
59	48, 53, 58	3eqtr4d 2814	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((exp‘𝐴)↑_𝑐(log‘𝑛)) = (𝑛↑_𝑐𝐴))
60	59	oveq2d 6808	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛))) = ((log‘𝑛) / (𝑛↑_𝑐𝐴)))
61	60	fveq2d 6336	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) = (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))))
62	61	breq1d 4794	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥 ↔ (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥))
63	39, 62	sylibd 229	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥))
64	63	expr 444	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → ((1 < 𝑛 ∧ (exp‘𝑧) < 𝑛) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
65	17, 64	sylbid 230	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
66	65	com23 86	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
67	66	ralrimdva 3117	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → ∀𝑛 ∈ ℝ⁺ (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
68		breq1 4787	. . . . . . . . 9 ⊢ (𝑦 = if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) → (𝑦 < 𝑛 ↔ if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛))
69	68	imbi1d 330	. . . . . . . 8 ⊢ (𝑦 = if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) → ((𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥) ↔ (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
70	69	ralbidv 3134	. . . . . . 7 ⊢ (𝑦 = if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) → (∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥) ↔ ∀𝑛 ∈ ℝ⁺ (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
71	70	rspcev 3458	. . . . . 6 ⊢ ((if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) ∈ ℝ ∧ ∀𝑛 ∈ ℝ⁺ (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥))
72	11, 67, 71	syl6an 655	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
73	72	rexlimdva 3178	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (∃𝑧 ∈ ℝ ∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
74	73	ralimdv 3111	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (∀𝑥 ∈ ℝ⁺ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
75		simpr 471	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ⁺)
76	1	adantr 466	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
77	76	rpefcld 15040	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (exp‘𝐴) ∈ ℝ⁺)
78		rpre 12041	. . . . . . . . 9 ⊢ (𝑚 ∈ ℝ⁺ → 𝑚 ∈ ℝ)
79	78	adantl 467	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ)
80	77, 79	rpcxpcld 24696	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((exp‘𝐴)↑_𝑐𝑚) ∈ ℝ⁺)
81	75, 80	rpdivcld 12091	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℝ⁺)
82	81	rpcnd 12076	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
83	82	ralrimiva 3114	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑚 ∈ ℝ⁺ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
84		rpssre 12045	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
85	84	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
86	83, 85	rlim0lt 14447	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥)))
87		relogcl 24542	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
88	87	adantl 467	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (log‘𝑛) ∈ ℝ)
89		simpr 471	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → 𝑛 ∈ ℝ⁺)
90	1	adantr 466	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
91	89, 90	rpcxpcld 24696	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
92	88, 91	rerpdivcld 12105	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℝ)
93	92	recnd 10269	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
94	93	ralrimiva 3114	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
95	94, 85	rlim0lt 14447	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
96	74, 86, 95	3imtr4d 283	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0 → (𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0))
97	6, 96	mpd 15	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1630 ∈ wcel 2144 ≠ wne 2942 ∀wral 3060 ∃wrex 3061 ⊆ wss 3721 ifcif 4223 class class class wbr 4784 ↦ cmpt 4861 ‘cfv 6031 (class class class)co 6792 ℂcc 10135 ℝcr 10136 0cc0 10137 1c1 10138 · cmul 10142 < clt 10275 ≤ cle 10276 / cdiv 10885 ℝ⁺crp 12034 abscabs 14181 ⇝_𝑟 crli 14423 expce 14997 logclog 24521 ↑_𝑐ccxp 24522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-inf2 8701 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 ax-addf 10216 ax-mulf 10217

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-iin 4655 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7043 df-om 7212 df-1st 7314 df-2nd 7315 df-supp 7446 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-2o 7713 df-oadd 7716 df-er 7895 df-map 8010 df-pm 8011 df-ixp 8062 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-fsupp 8431 df-fi 8472 df-sup 8503 df-inf 8504 df-oi 8570 df-card 8964 df-cda 9191 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-q 11991 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-ioc 12384 df-ico 12385 df-icc 12386 df-fz 12533 df-fzo 12673 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-fac 13264 df-bc 13293 df-hash 13321 df-shft 14014 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-limsup 14409 df-clim 14426 df-rlim 14427 df-sum 14624 df-ef 15003 df-sin 15005 df-cos 15006 df-pi 15008 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-ip 16166 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-hom 16173 df-cco 16174 df-rest 16290 df-topn 16291 df-0g 16309 df-gsum 16310 df-topgen 16311 df-pt 16312 df-prds 16315 df-xrs 16369 df-qtop 16374 df-imas 16375 df-xps 16377 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-mulg 17748 df-cntz 17956 df-cmn 18401 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-fbas 19957 df-fg 19958 df-cnfld 19961 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-nei 21122 df-lp 21160 df-perf 21161 df-cn 21251 df-cnp 21252 df-haus 21339 df-tx 21585 df-hmeo 21778 df-fil 21869 df-fm 21961 df-flim 21962 df-flf 21963 df-xms 22344 df-ms 22345 df-tms 22346 df-cncf 22900 df-limc 23849 df-dv 23850 df-log 24523 df-cxp 24524

This theorem is referenced by: cxploglim2 24925 logfacrlim 25169 chtppilimlem2 25383 chpchtlim 25388 dchrvmasumlema 25409 logdivsum 25442

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